Surds –

⚠ Common mistakes

• ✗Trying to add unlike surds. √3 + √5 ≠ √8. You can only add surds when the numbers under the root are identical.
• ✗Not fully simplifying. Writing √50 instead of 5√2 will cost you marks. Always check for the largest perfect square factor.
• ✗Forgetting to multiply BOTH numerator and denominator when rationalising. You must multiply top and bottom by the same expression to keep the fraction equivalent.
• ✗Using the wrong conjugate. For a denominator of (3 + √5), the conjugate is (3 − √5), not (−3 + √5) or (−3 − √5).
• ✗Leaving a surd in the denominator when the question says "rationalise". If the denominator still contains a root, you have not finished.

✦ Exam tips

• →Simplify surds as early as possible in a multi-step problem. Smaller numbers reduce calculation errors.
• →Know your perfect squares up to 225. This makes it quick to spot the largest square factor. See our formulas guide for values worth memorising.
• →Surds often combine with Pythagoras and trigonometry. If a right-angled triangle has sides of length √3 and √5, the hypotenuse is √(3 + 5) = √8 = 2√2. Leaving answers in surd form is usually expected.
• →Check the required form. If a question says "give your answer in the form a + b√c", make sure you present your answer exactly that way.